## Related questions with answers

**Reliability of a manufacturing network**. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $(x)$ of each line is shown below. Assume the lines operate independently. Find the standard deviation of the maximum capacity for each line. Then, give an interval for each line that will contain the maximum capacity with probability of at least .75.

Line | Maximum Capacity, $x$ | $p(x)$ |
---|---|---|

1 | 0 | $.01$ |

12 | $.02$ | |

24 | $.02$ | |

36 | $.95$ | |

2 | 0 | $.002$ |

35 | $.002$ | |

70 | $.996$ |

Solution

VerifiedWe need to determine the standard deviation and determine an interval containing the maximum capacity with at least a 25% chance.

**Chebyshev's Rule** tells us that:

- At least none of the data values are within one standard deviation of the mean.
- At least 3/4 or 75% of the data values are within two standard deviations of the mean.
- At least 8/9 or about 89% of the data values are within three standard deviations of the mean.

Thus the interval of interest will be two standard deviations about the mean.

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